Since #ln(t)->infty# as #t->infty#, it follows that #ln(ln(t))->infty# as #t->infty# (though very, very, slowly).

To illustrate how slowly #ln(ln(t))->infty# as #t->infty#, you might ask: how big should #t# be so that #ln(ln(t))>10#? (for example)

To answer this, solve the inequality #ln(ln(t))>10# by exponentiation: #ln(ln(t))>10 <=> ln(t)>e^(10) <=> t>e^(e^(10)) approx 9.4 times 10^(9565)#

In general, in order for #ln(ln(t))# to be bigger than an arbitrary #M>0#, you need to choose #t# so large that #t>e^(e^(M))#.